Check out the rest at:
I thought I would write a computer routine to check if two figures were congruent by the CCSS definition (rigid motions). One day I will post it.
The most important thing was to be specific as to what is a geometrical figure. You can read the CCSS document from front to back, back to front, upside down and more, but NO DEFINITION of a geometrical figure. For the computer program I decided that a geometrical figure was simply a set of points. My diagram may show some of them joined, but any two points describe a line segment (or a line). So a line segment “exists” for any pair of points.
The question is “Are the two figures shown below congruent or not?
I rest my case…..
Idly passing the time this morning I thought of a – b = a + (-b).
Fair enough, it is the interpretation of subtraction in the extended positive/negative number system.
I then thought of a – (b + c)
Sticking to the rules I got a + (-(b + c))
To proceed further I had to guess that -(b + c) = (-b) + (-c)
and then, quite ok, a – (b + c) = a – b – c
But -(b + c) = (-b) + (-c) is guesswork.
I cannot see a rule to apply to this situation.
The only way forward is to use -1 as a multiplier:
So a – b = a + (-1)b = a + (-b),
and then -(b + c) = (-1)(b + c) = (-1)b + (-1)c = (-b) + (-c)
by the distributive law.
It’s not surprising that kids have trouble with negative numbers!
Do we just assert that the distributive law applies everywhere, even when it is only defined with ++’s ?
George, to his teacher:
I have now integrated my preconceived ideas and the enlightenments engendered by yourself, but I still have trouble differentiating between “the limit of” and “the limits of”.
George’s teacher, aside:
I think George would be better off writing a novel. he could call it “The Limits of Continuity”.
I did the sums, no hesitation.
But then it asked for explanation.
“I know it’s right”, I wrote down fast,
“I understood from first to last!”.
“I’m going to be a mathematician,
“Not a fingernail technician!”.
Alfred North Whitehead, professor of mathematics and philosophy, and famous for his collaboration with Bertrand Russell on their joint effort, the Principia Mathematica, also wrote a book, “Introduction to Mathematics”, in 1911, for High School students and others who really wanted to know what math was all about.
The section on negative numbers is so relevant to the teaching of that topic today that it is a MUST READ. Click the link to download this section.
As h approaches zero
I quietly despair.
It really is the limit.
Please don’t take me there.
The funny thing about the calculus is that it was brought into existence by Isaac Newton in 1666 or earlier, and was developed and used without the idea of limits for over 150 years. The first attempt to get rid of the troublesome infinitesimals was by Cauchy in 1821, where he introduced the chord slope (f(x + h) – f(x))/h. The whole business of finding a satisfactory definition of the derivative was finally achieved by Weierstrass in the mid 19th century.
So here we go with cubics, and the same approach can be used for any whole number power of x, even negative ones. You should try it.
Next time sin(x) and cos(x), so no more sin(h)/h stuff.